The main advantage of a bull barrel is that it takes longer to heat up.

The additional weight will attenuate recoil. Even though a .223 really doesn't recoil that much, it will add up over time if you are in a target rich environment.

One problem with a bull barrel is that it screws up the rifle's balance. You may want to add weight to the butt stock to balance the rifle to your liking. However, more weight=less recoil.

What you will want is a free floating barrel. That is the one most important thing you can do to an AR to make it more accurate. At that point if you choose to add a heavier free floating barrel, it will be cheaper to simply buy another upper.